On q-Deformed Real Numbers

Abstract

We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a “q-analogue of a real.” The construction is based on the notion of q-deformed rational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain Laurent series.

Keywords:

• q-Rationals
• q-continued fractions
• total positivity
• Farey tessellation

Acknowledgements

We are grateful to Vladlen Timorin for a suggestion to look at Taylor coefficients. We are also grateful to Frédéric Chapoton for a SAGE program computing q-rationals and to Doron Zeilberger for the proof of the functional equations formulated as conjectures in the first version of this paper. This paper was partially supported by the ANR project SC3A, ANR-15-CE40-0004-01.

Notes

1 This observation is due to Doron Zeilberger.